In this paper, we study the asymptotic properties (bias, variance, mean squared error) of Bernstein estimators for cumulative distribution functions and density functions near and on the boundary of the $d$-dimensional simplex. The simplex is an important case as it is the natural domain of compositional data and has been neglected in the literature. Our results generalize those found in Leblanc (2012), who treated the case $d=1$, and complement the results from Ouimet (2020) in the interior of the simplex. Different parts of the boundary having different dimensions makes the analysis more difficult.

翻译：在本文中,我们研究了伯恩斯泰因测算员在美元-维简单x的附近和边界上累积分布函数和密度函数的无症状属性(比值、差异、平均平方差),这是一个重要的案例,因为它是组成数据的自然领域,在文献中被忽略。我们的结果概括了Leblanc(2012年)中处理案件的人($d=1美元),并补充了简单x内部Oimet(202020年)的结果。 边界的不同部分具有不同层面,使得分析更加困难。